The move would be part of a wider British bid for the tournament, which will have expanded to include 48 participating nations.
The former Scottish Labour leader has called on Nicola Sturgeon and the SFA to show “ambition” and give the game north of the Border a much- needed boost.
Hampden is due to host games as part of the Euro 2020 Championships, with Glasgow among 13 host cities from across Europe.
“People talk about ambition a lot, but they rarely indulge in it,” Mr McLeish told a Sunday newspaper.
“We might fail if we try this, but we need to forget that and give it a try because ambition will make a country and for football this can be our time if we chose to strive.
“It can’t be a matter of making noises from the sidelines. If we are the country we think we are, we should be boldly pursuing the goal.
“There is no reason why Scotland shouldn’t lead a British bid to host the 2030 World Cup.
“Nicola Sturgeon needs to rise to the challenge, show serious intent, and make this happen along with the SFA. We have been in an abysmal place since we last qualified for the World Cup in 1998.”
The SFA said it would not rule out a joint bid with the other nations of the UK, but no talks have taken place.
Scotland was previously part of a joint bid with Ireland to host the 2008 European Championships, but lost out to another joint bid from Switzerland and Austria.
Mr McLeish urged the SNP to set aside any concerns that the prospect of a joint UK bid could raise over the party’s longer term political goal of an independent Scotland.
“If we are still part of the UK then it is not a problem, and if we are an independent country then it is still not a problem,” he said.
“The US, Canada, and Mexico will host the World Cup in 2026, so there are plenty of examples of different countries coming together to host.”
The Scottish Government said a bid for the World Cup was “a matter for the football authorities”, but added: “We would consider any proposal seriously should one come forward.”
The SFA said the focus was on Scotland’s involvement in Euro 2020 – which will be hosted across 12 countries – but added: “We are always open-minded to the prospect of hosting major tournaments.”